DISPERSION AND RESOLVING POWER OF PRISMS Dispersion.--The dispersion at any point in the spectrum is de fined as the rate of change of the angle of deviation with respect to a small change in the wave-length. Thus, if 0 be the angle of deviation from the original path of the light, the dispersion is represented mathematically by de/dX. Since 0 varies with the refractive index of the glass (a), and the refractive index varies with the wave-length (X), we may write Dispersion = dO/dX=d0/d,u.dy/dX.
Dispersion thus depends upon two factors. The second, dp./dX, depends solely upon the optical properties of the glass of which the prism is composed and may be calculated from observations of the refractive indices for different wave-lengths, when ex pressed in a formula. If the refractive indices be approximately represented by an equation of the form µ = a+ , the constants a and b being deduced from measured values of /4 for different wave-lengths, dp./dX= —2b/V. The negative sign merely im plies increase of p, with decrease of X, and it follows from the formula that the dispersive power varies roughly in proportion to the inverse cube of the wave-length. For example, the disper sion at X3,5oo in the near ultra-violet is about eight times that at X7,000 in the extreme red.
The other factor in the expression for the dispersion of a prism, dO/d,u, depends upon the angle of the prism, and on the way in which it is presented to the incident light. If, as in fig. i I a, the angles of incidence and emergence be represented by and the internal angles of refraction by and the angle of the prism by a, and the angle of deviation by 0, it is readily deduced from the fundamental relation sin i= it sin r that cos cos or, since dO = and a, de since = cos cos i2 This expression clearly shows the dependence of the dispersion on the angle of the prism and the angle of incidence, but, following Lord Rayleigh, a very convenient geometrical equivalent can be deduced; namely, d t2 dy a where and are the lengths in the prism traversed by the ex treme rays, and a is the breadth of the emergent beam. When the prism is at minimum deviation and the beam is large enough to cover it, is equal to the length of base of the prism. The whole dispersion may now be written, d 0 t2— = = • — • dX a dX The same expression can also be deduced directly by an applica tion of Fermat's principle of optical distances.
When there is more than one prism, the sum of the values of is to be taken, and in direct vision of compound prisms, the algebraical sum is taken, the crown glass prisms being regarded as negative.
Resolving Power.—The power of separating closely adjacent spectrum lines is called the resolving power of the prism. (See LIGHT.) As now employed this is a theoretical quantity, repre senting the separating power when the slit is indefinitely nar row, and is given numerically by R = X /dX, where dX is the differ ence of wave-lengths of two lines which can just be divided. R does not increase indefinitely as the slit is narrowed, because the image of the slit is broadened by diffraction. The actual image is a diffraction pattern, as roughly shown, with the corresponding intensity curve in fig. 12. Most of the light is concentrated in the central band, and the secondary bands can be disregarded. If f be the focal length of the camera lens, and a the effective breadth of the beam passing through it (the effective beam being sup posed of rectangular cross section) the half-breadth (p) of the central band is given by the elementary theory of diffraction as p= fX/a. It is to be observed that the shorter the wave-length the narrower is the band, while a reduction of aperture is accom panied by a broadening.
The condition for resolution may be conveniently explained by reference to fig. 13, where the spectrum is supposed to be thrown on a screen by the second lens. The wave-lengths of two closely adjacent lines from light entering the slit at s being X and X —dX, the corresponding separation by the prism is de. Each line ap pears as a diffraction band, of which the intensity curves are shown, and resolution is just effected when the maximum of one band falls on the minimum of the other, that is, when the distance between the centres of the two bands is p. Under these conditions the integration of the two bands gives a curve with two maxima and if the intensity of these be taken as unity, that of the mini mum between is o•8. Experience has shown that this difference is sufficient to resolve lines, and the condition for resolution is thus de= a; or, X= a.d0. In this expression, it should be remembered, a is not necessarily the diameter of the image lens, but the effective breadth of the beam which enters it. By definition, R= X/dX and we thus find R= a • dO/dX, or in Ray leigh's form, R= • dm./ dX.